[EM] Multiwinner Bucklin - proportional, summable (n^3), monotonic (if fully-enough ranked)

[Abd ul-Rahman Lomax]

At 10:59 AM 4/2/2010, C.Benham wrote:

>Unfortunately these top-two runoff versions break MCA's compliance 
>with Favorite Betrayal and Mono-raise.
>
>Top-rating your favourite F could cause F to displace your 
>compromise C in the runoff with your greater-evil E, and then F loses
>to E when C would have beaten E.

The problem arises with candidate elimination, but the analysis is 
incompletely. Sure, if all you do is top-rate F, bullet-voting, your 
vote could have this effect, with a two-candidate restriction. 
However, two candidates is an artificial restriction; some runoff 
systems use three, and repeated ballot, one of the most-used voting 
systems in the world (vote-for-one ballot, majority required), has no 
such problem, relying upon the desire of partisans to complete the 
election. If a majority do not want to complete the election, there 
you have it. Majority rule.

Repeated ballot, majority rule, using approval ballots, suggests a 
very obvious strategy, and, as Robert's Rules of Order points out, 
each pool is informed by the results of the poll before. The first 
poll will tend strongly to bullet votes for voters with a significant 
favorite. Then the approval cutoff is raised.

Bucklin collapses the first two or three rounds of this into a single ballot.

So, in the scenario described, the voter has voted a poor strategy, 
and sees a bad result. TANSTAAFL. If the voter is concerned about 
that risk, the voter should certainly have added an approval for C. 
Favorite Betrayal assume that the favorite is raised above the 
compromise, which causes the voter to lose utility, but in Bucklin 
methods, the preference, before the end of counting, i.e., before any 
runoff choice is made, is not maintained, it is reduced to equal 
ranking. Bucklin is an Approval method, simply one that allows 
preference to be first asserted before being collapsed to equal ranking.

>Also, like plain Approval followed by a runoff between the two most 
>approved candidates, it is *very* vulnerable to "turkey-raising"
>Push-over strategy. Voters who are fairly confident that their 
>favourite can get into the final runoff have an incentive to also approve
>(or top-rate, depending on the version) all the candidates they are 
>confident their favourite can pairwise beat in the runoff.

It's tricky. All candidate elimination schemes are vulnerable to 
possible turkey-raising, but as the number of preserved candidates 
increases, it becomes more difficult. Candidate elimination is the 
culprit, period, and it's worst when the restriction is to two. This 
is a reason why I strongly suggest allowing at least one additional 
write-in approval in a runoff. This makes a two-round system a closer 
simulation of repeated ballot.

It is possible, even, that the second round has no eliminations at 
all, other than withdrawals. Same ballot, but printed on the ballot 
are the results from the first round. Any significant write-ins are 
reported specifically. A new write-in is possible.

The second ballot is then pure Bucklin, no runoff, plurality wins 
after all rounds are collapsed if no majority found in the process.

What I strongly prefer to see is experimentation with Bucklin. We 
need more data to understand how the method works in reality, we do 
have historicasl data that indicates it worked well. (The common 
assertion is that, in some circumstances, only a small percentage of 
voters added additional approvals, but these were, I believe, primary 
elections, and first preferences will stand out even more in the 
minds of the voters. In the Bucklin elections where I've seen actual 
round data, there was hefty voting in the lower ranks. That in a 
one-round Bucklin system most voters, stable conditions -- not the 
first election, but after voters become accustomed to it -- may 
bullet vote, is not a failure of the system, rather it shows 
preference strength for the favorite.)

>Also some people might object that parties that run a pair of clones 
>have an advantage over parties that run a single candidate.

Of course. But this doesn't work if the second ballot isn't 
restricted. And it runs into the reality of political election 
process: two candidates means divided attention and campaigning, two 
names to raise in familiarity and approval, not just one. It can 
seriously backfire.

What seems clear to me is that a runoff system using Bucklin-ER, 
three-rank, for both ballots and a liberal inclusion in the second 
poll, incentivizes voting a true, sincere Range ballot, particularly 
in the first poll. "Approval" is tied to the utility to the voter of 
avoiding a runoff. If the voter prefers a runoff to any result that 
does not elect the favorite, the voter will bullet vote, a sincere 
and accurate vote.

The Range Ballot for three-rank Bucklin is one with ratings of 4, 3, 
2, 0, with 0 being the default. The rating of 2 is the rating of 
"equal or better than the expected result." In the second round, the 
voter has better knowledge of the expected result. It appears to me 
that optimal voting strategy in the first round is to sincerely 
apportion the candidates in the approved ranks according to perceived 
utility, and the existence of an additional round allows the voter to 
set the approval cutoff (<2) more conservatively or even absolutely 
as anything below the favorite.

In the second round, if it comes to that, the voter has an incentive 
to raise the approval cutoff, and voting becomes more "strategic," 
but only in the sense allowed by Approval and Range voting, i.e., a 
judgment of election probabilities becomes important, because if the 
voter does not approve a frontrunner, in the lowest rank, at least, 
the vote is likely wasted. And in a second round, the voter has no 
excuse for not knowing who the frontrunners are.

Indeed, top-two runoff is a method of providing that information. 
Write-in votes should still be allowed, and using a better system 
than vote-for-one in the runoff as well, allows this without creating 
a spoiler effect from a write-in candidacy. Elections have occurred 
where a write-in won a runoff election, and San Francisco, by 
outlawing, for the last election held by top-two runoff before IRV, 
write-in votes, on the argument that it might not result in a 
"majority," and that voters had been promised such in the measure 
implementing top-two, quite possibly avoided such a victory. If a 
write-in candidate wins a plurality in a runoff election, we can be 
about positive that this was the best result! Indeed, we should make 
it easier, not harder.

Ironically, that same promise was made to voters with the measure 
that implemented IRV, explicitly ("majority of the votes"), and the 
system, as expected by anyone who understood, fails to accomplish it, 
whereas top-two runoff almost always did. Politics. Sucks. I'm 
disgusted by public officials who manipulate voting systems for 
partisan benefit. They are the enemies of democracy, and, by ignoring 
substance and focusing on a very shaky interpretation -- 
preposterous, actually -- of what an "election" is, the California 
Supreme Court confirmed San Francisco's position. And where were the 
voting systems experts? I'll tell you: gazing at the navel of voting 
systems criteria instead of what is actually the best voting system 
in common use, top-two runoff, which could almost trivially be 
optimized to avoid its big failure, shared with IRV, Center Squeeze, 
another way of looking at Favorite Betrayal.

>*Voters give each candidate a Top, Middle or Bottom rating. Default 
>rating is Bottom.
>If one candidate X  (based on these maybe constrained 
>ballots)  pairwise beats all others, elect X.
>Otherwise, interpreting Top and Middle rating as approval, elect the 
>most approved candidate.*

Let's put it this way: if a Condorcet winner can be found in the 
ballots, that candidate should definitely either win or be in a 
runoff. Much hot air is expelled over the issue of multiple 
majorities, where a Condorcet winner may be less approved. It's an 
unlikely scenario, in general, the tendency in approval systems is 
toward bullet voting and majority failure, not toward multiple 
majorities. But I'd still want to see a Condorcet winner in a runoff, 
even if not in the top two by approval. Rare, this would be. Possibly 
so rare and Condorcet winner by such a small margin, with such a 
small loss of utility by choosing, instead, the most-approved, that 
it would not be worth the extra trouble of ballot analysis.

In any case, a science, like election science, should be based on 
study of actual function and evidence, not merely on theoretical 
analysis, which can easily place too much weight on criteria of 
speculative or conditional value.

In theoretically studying conformance to preference-based criteria 
that give no importance to preference strength, students and experts 
have failed to study how real voter utilities will motivate voter behavior.

If a voter has high preference for a set of two candidates over all 
others, and the less-preferred is only lesser by a weak preference, 
suppressing the preference is incentivized. That's not Favorite 
Betrayal, though, it is something else, and I don't know a handy name 
for it, it is Not Expressing A Known Preference. This particular form 
of strategy, though, doesn't seem to be at all necessary in Bucklin, 
because of the allowed expression that later becomes equal ranking.